×

The algebra of integro-differential operators on an affine line and its modules. (English) Zbl 1272.16027

For a field \(k\) of characteristic zero denote by \(\mathbb I_1\) the algebra of linear operators on the polynomial ring \(k[x]\) in one variable \(x\), which is generated by the operator of multiplication by \(x\), by the operator \(\tfrac{d}{dx}\) and by the operator \(\int\) such that \(\int x=\tfrac{x^n}{n+1}\). It is shown that \(\mathbb I_1\) admits a \(\mathbb Z\)-grading. There is given a classification of all irreducible \(\mathbb I_1\)-modules. If \(a\) is a nonzero element of \(\mathbb I_1\) and \(M\) is a left \(\mathbb I_1\)-module of finite length, then the map \(x\mapsto ax\) on \(M\) has finite dimensional kernel if and only if it has a finite dimensional cokernel. For an irreducible module \(M\) there is given a criterion under which \(aM\) has finite dimension. It is proved that \(\mathbb I_1\) is a coherent algebra in which every finitely generated left ideal is two-generated. Every left (right) ideal of finite length is cyclic. For a generic element \(a\in\mathbb I_1\) its centralizer is a finitely generated \(k[a]\)-module of rank \(>1\). There are found all elements \(a\in\mathbb I_1\) for which \(ak[x]=k[x]\) and for which the map \(f\mapsto af\) is injective on \(k[x]\). One-sided invertible elements of \(\mathbb I_1\) are also considered.

MSC:

16S32 Rings of differential operators (associative algebraic aspects)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amitsur, S. A., Commitative linear differential operators, Pacific J. Math., 8, 1-10 (1958) · Zbl 0218.12054
[2] Artamonov, V. A.; Cohn, P. M., The skew field of rational functions on the quantum plane, J. Math. Sci. (New York), 93, 6, 824-829 (1999) · Zbl 0928.16029
[3] Baer, R., Inverses and zero-divisors, Bull. Amer. Math. Soc., 48, 630-638 (1942) · Zbl 0060.07103
[4] Bavula, V. V., Generalized Weyl algebras and their representations, Algebra i Analiz, 4, 1, 75-97 (1992), English transl. in St. Petersburg Math. J.4 (1993) no. 1, 71-92 · Zbl 0807.16027
[5] Bavula, V. V., Simple \(D [ X , Y ; \sigma , a ]\)-modules, Ukrainian Math. J., 44, 12, 1500-1511 (1992) · Zbl 0810.16003
[6] V.V. Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, in: Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), 23 pp., Carleton-Ottawa Math. Lecture Note Ser., 14, Carleton Univ., Ottawa, ON, 1992. · Zbl 0802.17006
[7] Bavula, V. V.; van Oystaeyen, F., The simple modules of certain generalized crossed products, J. Algebra, 194, 2, 521-566 (1997) · Zbl 0927.16002
[8] Bavula, V. V., Structure of maximal commutative subalgebras of the Weyl algebra, Ann. Univ. Ferrara-Sez. VII-Sc. Mat., LI, 1-14 (2005) · Zbl 1120.16022
[9] Bavula, V. V., Dixmier’s problem 5 for the Weyl algebra, J. Algebra, 283, 2, 604-621 (2005) · Zbl 1069.16031
[10] Bavula, V. V., Dixmier’s problem 6 for the Weyl algebra (the generic type problem), Comm. Algebra, 34, 4, 1381-1406 (2006) · Zbl 1094.16016
[11] Bavula, V. V., Module structure of the tensor product of simple algebras of Krull dimension \(1\), Representation Theory of Groups, Algebras, and Orders (Constanta, 1995). Representation Theory of Groups, Algebras, and Orders (Constanta, 1995), An. Stiint. Univ. Ovidius Constanta Ser. Mat., 4, 2, 7-21 (1996) · Zbl 0880.16003
[12] Bavula, V. V., The algebra of integro-differential operators on a polynomial algebra, J. Lond. Math. Soc., 83, 2, 517-543 (2011) · Zbl 1225.16010
[13] Bavula, V. V., The group of automorphisms of the algebra of polynomial integro-differential operators, J. Algebra, 348, 233-263 (2011) · Zbl 1258.16039
[14] V.V. Bavula, An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators, J. Algebra (2012) (to appear). · Zbl 1278.16024
[15] V.V. Bavula, The largest left quotient ring of a ring. Arxiv:math.RA:1101.5107. · Zbl 1353.16037
[16] Berest, Yu; Wilson, G., Mad subalgebras of rings of differential operators on curves, Adv. Math., 212, 1, 163-190 (2007) · Zbl 1119.16024
[17] Block, R. E., Classification of the irreducible representations of \(s l ( 2 , C )\), Bull. Amer. Math. Soc., 1, 247-250 (1979) · Zbl 0412.17009
[18] Block, R. E., The irreducible representations of the Lie algebra \(s l ( 2 )\) and of the Weyl algebra, Adv. Math., 39, 69-110 (1981) · Zbl 0454.17005
[19] Burchnall, J. L.; Chaundy, T. W., Commutative ordinary differential operators, Proc. Lond. Math. Soc., 21, 420-440 (1923) · JFM 49.0311.03
[20] Dixmier, J., Sur les algèbres de Weyl, Bull. Soc. Math. France, 96, 209-242 (1968) · Zbl 0165.04901
[21] Jacobson, N., Some remarks on one-sided inverses, Proc. Amer. Math. Soc., 1, 352-355 (1950) · Zbl 0037.15901
[22] Goodearl, K. R., Centralizers in differential, pseudodifferential, and fractional differential operators rings, Rocky Mountain J. Math., 13, 4, 573-618 (1983) · Zbl 0532.16002
[23] Hellström, L.; Silvestrov, S., Ergodipotent maps and commutativity of elements in non-commutative rings and algebras with twisted intertwining, J. Algebra, 314, 17-41 (2007) · Zbl 1154.16023
[24] Mazorchuk, V., A note on centralizers in \(q\)-deformed Heisenberg algebras, AMA Algebra Montp. Announc., Paper 2, 6 (2001) · Zbl 0981.16025
[25] McConnell, J. C.; Robson, J. C., Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra, 26, 319-342 (1973) · Zbl 0266.16031
[26] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian Rings (1987), Wiley: Wiley Chichester · Zbl 0644.16008
[27] Stenström, B., Rings of Quotients. An Introduction to Methods of Ring Theory (1975), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0296.16001
[28] Stafford, J. T., Module structure of Weyl algebras, J. London Math. Soc., 18, 3, 429-442 (1978) · Zbl 0394.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.